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Plane problem of elasticity theory in a multiconnbcted domain
UDC 539.3
PLANE PROBLEM OF PLASTICITY
THEORY XN A MULTICONNECTED
PMM Vol.43,
No.5, 1979, pp. 846-858 A. I. KALANDIIA (Tbilisi) (Received November 20, 1978)
Fredholm integral equations are constructed for unknown normal stresses on the domain boundary in the fundamental problem of plane elasticity theory, and their investigation is given. l. Formulation of the problem.Weshallsolvethefundamental plane problem of elasticity theory for external forces given on the boundary when the elastic medium occupies a finite or infinite domain of the plane of the complex variable z = 2 + iy, and the external forces on each of the bounding domains of the closed contours are statically equivalent to zero. We consider the (connected) domain S7 bounded by simple closed non-intersecting contours L,, L,,. . . , L,, of which the first encloses all the rest. The symbol L will denote the set of all contours. There may be no external contour L, , and then the domain s+ will be infinite (plane with holes). Let s- denote the domain complementing s+ in the whole plane, Sk- the domain enclosed within Lk (k = 1. 3.. . .’ in), and SD+, s,the domains inside and outside of L, , respectively. Furthermore, let n *be the normal to the line L drawn at any point and external relative to St, and T the direction of the positive tangent to - L at the same point. For definiteness, we consider that direction on I, positive which keeps the domain St on the right. We assume the line L to have continuous curvature everywhere. 2. On properties of stress functions in a multiconnected domain, Representation of biharmonic fun c t i o n s, To facilitate the presentation let us recall certain known formulas of plane elasticity theory and let us indicate the properties of the stress functions. We borrow all the necessary formulas from the book by N. I. Muskhelishvili [l]. The biharmonic stress function IL(7, g) (Airy function) and its first derivatives are representable by the formulas u = Re [Z(F (z) -I- y. (r.)], Z = .Z - iy (2.1) &L i)rr $ (2) -= j(’ (r.) ClX -+ i dy xzz‘p (3) + ZG) + 9(z), where cp, $, x are analytic functions of the variable 2 in a domain occupied by S+ they have the form (see [l], Sect. 35) an elastic medium. In the finite domain 9 (z) = z 5 k-1
A, In
(2.2) (
k=l
910
911
Plane problem of elastlclty theory
x (2) =
2
yA.’ In (2
i
k=l
‘II (‘1 = i
k=l
yk’ln
tz-
zk) + $,* tz)
where zk is a certain fixed point within Sk- (k = 1, 2,. . ., m), Ak is an arbitrary real number, ykr Yk’, YkH are arbitrary complex constants, and (Fe1 xer *,* are holomorphic (single- valued, analytic) functions in S +. It is seen that for the Airy function to be single-valued in S+ , it is necessary and sufficient to comply with the conditions Bk = yk’, Im yk” = 0 (k = 1, 2,. . .,
m)
(2.3)
and this means (see [l], Sect, 33) that the external forces on the contour of each of the holes are statically equivalent to zero. In the presence of (2.3), for the elastic displacements to be single-valued it is necessary (and sufficient) that Al, = 0, yk = 0 (k = 1, 2,.. ., m) and this is equivalent to the requirement of holomorphicity of ‘p (z) in latter conditions, expressed in terms of the function u (5, Y), are
s
f3AU ~cls=O,
’ St
6Au xan-Auan
cls=O
a2 > (k=l,2,
ds=O
S+. These
(2.4)
. . ..m)
S+ , Therefore, if the stress function u (z, Y) is single-valued in the domain then for the corresponding elastic displacements to be single-valued, it isnecessary and sufficient to comply with the conditions (2.4). ‘p (z), 9 (z) are holomorphic Upon compliance with these conditions, the functions x (z) will generally be multivalued (it has the form of the in SC, and the function right side in the second relationship of (2.2) for ok’ = 0, Im yh” = 0). The above-mentioned properties of the stress function were apparently first established by GrioIi [2] on the basis of the paper [S]. &, , Sf is an infinite domain located outIn the case when there is no contour In this case, under the same assumptions relative side the contours L,, L,,.. ., L,,. and the single-valuedness of the displacements, to the external forces applied to Lk , the nature of the stress functions as well as the potentials cp, x, ‘p remains as before in any finite subdomain of the domain S+. However, the functions cp and x will not generally be holomorphic in the whole domain S+. It is known that the Kolosov - Muskhelishvili potentials admit of the following representations in the neighborhood of the infinitely distant point (see [l], Sect. 36) (2.5) cp (z) = ‘Fo (z) + cp* (Z)? x (z) = X0 (z) + X* (2) (p. (z) = rz, x0 (z) = I”z2, I” = B' + iC'
A. I. Kalandiia
912
Here 1’, B’, C’ are real constants characterizing the given stress field at infinity, and 3. is a certain real constant. The formulas (2.6) mean that the functions ‘p* (z), are holomorphic outside a circle L, of sufficiently large radius R with 9* (z) center at the origin, including the infinitely distant point. Let us present certain fundamental formulas for representations of the biharmonic functions which are derivable by a standard method by using appropriate elementary solutions. A certain difficulity occurs just in the case of the infinite domains when the behavior of the function under consideration must be taken into account in infinitely remote parts of the plane. For the class of functions which we shall discuss later, this behavior is characterized completely by (2.5). The letters P, P, wiLl denote points of the domain S+ or S- , and Q, PO will denote points on the line I;, where Q will usually be the variable of integration. The affixes of these points on the complex plane shall be Z, zlr t, to, respectivz = .Z + iy, z1 = x1 + iy,. The arc abscissas of the points t, to are ely, where denoted by s and so and are measured in the direction of the positive tangent T to the line L on each contour L, (k = 0, 1, . . ., m) . We shall not introduce new symbols to denote the coordinates of the points Q and PO by assuming t = 5 -t iy, to = x0 -;- iy,, or more accurately, t = X(S) + my(s) and to = z (so) f iy (so). To avoid the introduction of new symbols for the functions f of points of the line L we shall also sometimes not distinguish the notation f (t) and f (Q) or f (to) and f (I’,). Writing the Green’s identity for the pair of functions u, Au, and then for Au, 0, adding the formulas obtained, and relying upon the elementary solution of the biharmonic equation, we obtain the known representations for a finite domain 1
5
P;
[LO (u;
u (a = -yjy
Q) -i1 (u; P; Q)] dsy
Z’ E ,j”
(2.7)
L
0 =
& s’ [Lo (u;
P;
Q) + 1 (n;
P;
PES-
Q)] ds,
L
Here the point
(a/an, 0): Pi
LO+; 1 (u; P;
Applying
is differentiation
Q) = 4 Q).+
1 [
r21n~
with respect to the normal direction
a
1 aAlL
--anq
QEL,
the Laplace operator to both sides of (2.7), P, Q) as,
L o=&
s
L
l(A.u; P, Q)ds,
PES-
at
I
A”anqr21nF
Q
& 11 (Au;
~0
1
+--l)~--ua$+,
A.u (P) =
to
r
= i5-tl
we obtain
P ES+
(2.8)
Plane problemof elasticity theory
The unction the condition
Au is evidently
harmonic
in
913
S+ , and should co~equently
satisfy
s
8Au YJ-+s=O
l
(2.9)
L
The known relationships
.(P)~ql(u:
s
o=$-
P. Q)ds,
PES-
P, Q)b,
l(u;
(2.10)
PEF
L
follow from (2.7) for a function u (x, Y) which is harmonic in S . Let us examine the case of an infinite domain in greater detail. Let us introduce a domain Sn of radius LR
L,, L,, . . ., ,!I+,, and within the circle is taken so large that the circle L, would enclose all the contours L,, k = 1, 2,. . ., m, whose set, i.e., total domain boundary, will again be denoted by L. Let us set located outside the contours R with center at the origin;
Uwhere
uo
R
(2.11)
ug = u+
is defined by the first formula in ( 2.1) and the last three formulas in (2.5) uO = Re [Z cpo(4 +
x0
(41
(2.12)
=
ps [r --f-B’ cos 26 - C’ sin 2 61, z = Wie Gxfm) = 2 (f - B’), q$(=) = 2 (r + B’), TXJrn) = 2 C’ (a,, uV, zxV are the stress components). components, i. e., the quantities (2.13),
(2.13)
Let us recall that the values of the stress
are given at infinity when considering the plane problem for domains containing the infinitely distant point of the plane. The right side of (2,12) is an Airy function corresponding to the homogeneous stress field caused by the forces (2.13). It is biharmonic in any finite part of the plane and for large 1z 1 ug = a (p*) (2.14) The relationships (2.7) are evidently valid in S,
for the function
1~-uo . There-
fore 2 u* (P) = 2n
s
[r,” (u,;
P,
Q, + 2 (u*:
P,
Q)] ds,
P ES,
L+Lf$ 0
1 =-zi-
s L+LR
[L” (u*;
P,
Q) + 1 (IA,; P,
Q)]ds,
P E
Sk-,
k = 1,x,
... , m
Moreover, analogously to (2.9)
s
L-tLR
Let us consider the function
aAu, an
(2.15) ds=O
A. L Kalandiia
914
(2.16) 1 I(Au,;P, Q)ds
Ax0 (P) = -&
“k
where P is an arbitrary point in any finite part of the plane first formula in (2.6), the estimates
i-‘Au,
Au = 0 (I z I-“), are valid for large I z ‘I s consequently It hence follows that
in the limit
as
that upon com$ance
cannot grow more rapidly than plnp hence conclude that for large 1z )
on the plane
z
30.
(2.17)
(2.15) becomes
’ r3Air a,2 as = 0
(2.18)
with the condition
=
tends to zero at H -
\ i,
at infinity.
Lao
Now on the basis of (2.17)
in (2.16)
XI , the equality
ds =
on the basis of the
0 (I I” I-3)
0 everywhere
R -
- rlh, 7 s O/I I, It is almost evident
=
the integral
A(/%,On the same basis,
an
z.
!I
(2.18)
the function
Taking (2.14) into account,
(I”“)
we
(2.19)
and (2.19) 00 (I. Y) = 31 + BY + Y
follows, where a, B, Y are arbitrary real constants. this trinomial since it does not yield nonzero stresses. In the long run, the representations 1 ’ 7~(P) = u,, (PI -kg-
Q) T I (I!.; P,
Q)] ds,
[L” (u; I’, Q) + 1 (!I; f’, Q)] ds,
PfSh_-,
s
[L” (1~; P,
We shall not turn attention
to
(2.20)
P E S+
L ~0
(p) =
-&,\
h-=1,2,
,..
, m
L
hold for the stress function Correspondingly
u (x, y) in the infinite 1
domain
S+
(2.21)
Au0 (Pj = -&
1 1 (Au; P, L
Q) ds,
PESk-,
k=l,2,
.... m
Planeproblem of elasllcity theory
915
Now (2.18) should still be added to the preceding formulas, It is understood the conditions (2.4) that the displacements be single- valued remain unchanged. 3. Integral equations of the conditions introduced, the plane problem generalized plane state of stress of an elastic stress function u (x, y) in the domain S+ g+
ig
that
the plane problem. Under (the problem of plane strain or the body) is to determine the biharmonic by means of the boundary conditions
=fl(t)+if2(t)fc(t)
on
L
(3.1)
where fl, f2 are functions given on L which are single-valued and continuous each of the contours Lk comprising L , and subject to the conditions:
s
flds+f:!dy=O
on
(k=1,2,...,7?2)
(3.2)
and c (1) - ck = ak + iflk on Lk, where ck are complex constants to be determined together with the function u . Only one of the constants ck can be fixed arbitrarily; we shall consider c (t) ‘= 0 on Lo . Moreover,. it is required from the function u that it satisfy the following additional conditions
s.
J$is=O
(k=
0,1,2,...,m)
(3.3)
9;
SC
a$-Au$)ds=O,
~(y~-Au~)&~O
LI;
(3.4)
Lk
(k = 1,2, . . . ) m)
In the case of an infinite domain, the boundary conditions (3.1) are given in the set L,, L?,. . ., L,, and in conformity with this, one of the additional conditions, namely (3.3) ) drops out for k = 0 . The condition at infinity u(z,
Y) = UO(T,
Y) iO(lzl)
(3.5)
should still be appended to these conditions, where ~0 is given by (2.12). and the second term is the right side of the first formula in (2.1) if the corresponding right all the second sides of (2.6) are substituted in place of cp and x . (More accurately, order partial derivatives of u should be given at infinity, and this wilI be equivalent to giving the function ?.Aitself in the form (3.5) ). The COIBtant dk can here alS0 be given arbitrarily on any one of the contours Lk (k = 1, 2,. . . , m), be considered zero, say, as we shall do. Therefore, the number of unknown real constants to be determined during the solution of the problem equals 2m and 2m - 2 in the case of the finite and infinite domains, respectively. proceeding to the construction of the integral equations for the finite domain S’J we rewrite condition (3.1) in the form ~=go(t)+%z-!-BkY+~k
=s
(3.6)
s
l
go(t)
j&x t- fz4/,
0
b(t)
=
f,$ -+j,:
on Lk(k = 0, I,..., m)
916
A. L Kalandiia
According to the above, it can be considered neglect the constants aR since they play no part Let us insert (3.6) in place of 11 and &di3n ity in (2.7), and let us take into account that the everywhere harmonic. Then by using (2.10) we entation for the solution of the problem formulated ted in the right side): (A(P),
-&\
?U(P)-=
&S
’ L (u; P; Q)ds
that o,,, =_ f10 mmOY and we shall in solving the problem. in the right side of the first equalfunction ahx -t- Fky + hk is obtain the following integral repres(the additive constant X0 is omit-
‘; w(P),
PES’
(3.7)
[(lllf-l)h”~V--p,(Q)~~,lf]ds
(3.8)
The function zu (P) (the sum of simple and double layer potentials with known densities) is given on the whole plane. It undergoes a discontinuity when the point P passes through the line L t which is not essential to the subsequent exposition. Let us pass to the limit P -+ P o (PO E L) in (3.7), and let us differentiate the limit equality twice with respect to the contour arc. (For the differentiation with respect to the arc s to be legitimate, it is necessary that the given functions go and h, be subject to definite smootheness conditions which are easily refined). Then using the boundary condition (3.6) for the function u (P) desired, we find (the prime denotes differentiation with respect to s)
-s’ -f$ 1
2x
L” (u; PO, Q) ds = g (to) + QX~ + pkq/
on
L,,
(3.9)
L
(k = 0, 1,2:.
. . , m;
g (to) = g,” (to) -
t” = .ql -.I- iy,) a2w (PO) 1 &2
(3.10)
We also pass to the limit as P 3 PO in the first equality in (2.8) and we use Then the known formula for the limit values of a double-layer potential. u; Pm Q)ds,
Au(P,,)=$~E(A
P,, E
L
(3.11)
L
where the limit values on the line L for the function Au are in the left side and values of the integral in the first equality in (2.8) on the same line are in the right side, The set of equations (3.9X (3.11) yields a system of integral equations in the unknown boundary values of the functions AU and dAu / dn. After the calculaJ_,” under the integral sign in (3.9), tions related to differentiation of the operator by using the notation Au(Q) we write the system of integral &
=; o CC% equations
+Au
= v(Q)
in the form
1 [kir (to, t) v (r) + k,, (to, r) o (t)] ds = g (to) + or&” + l%vo” L th(to, s
qv
(t) -t
(3.12)
h27,(to, 0 CJ@>I d.9 = 0
(3.13)
Plane problem of elrutldty
-
4k,, (to, t) = 2 (hl r + 1) + k (to)
-
COS
917
theory
2a (t, t,,) -
r (21n F
+
i)X
sin a (t, tJ
2k,, (to, t) = r-l sin a (t,, t) - r-l sin 2a (t, t,) cos a (to, t) k (to) sin a (t, to) sin a (to, t) + k (to) cos ]a (t) - a (t,)l x
(In r + l/J kzl (to, t) = 2 (In F + I),
kaz(to, t) = 2 $--In
+
=
Q 2r-‘sin
a (to, t)
(F =
1t -
to ( )
where k (t) is the curvature of the line L at the point t, a (t) is the angle formOT, ed by the (positive) tangent to the line L at the point t and the axis and a (to, t) is also an angle formed by the same tangent and the vector t,t , and a (t,) and a (t, t,) measured in the positive direction from this latter. The notation has an analogous meaning (the angles a (t,, t), and a (t, to) are shown graphically with their measurement directions in [4] ). The additional conditions (3.3) and (3.4), which become
[v(t)ds=O
(3.14)
(k=O,l,...,m)
L;, ~[XV(t)+y’~(t)]ds=o, ;;=
S[yv(t)-.&o(t)]&-0 5
l,Z,....m)
in the new notation, are appended to (3.13). Let us mention still another relationship which the functions Y and o should satisty. To this end, we calculate the normal derivative of the desired function u (z, y) L . Taking into account that by means of (3.7) and we integrate it along the line the function 11)is harmonic in Sf , we find
s
&s
;ds=
I(t)=
b(t) Au
[a(t)%-
I
ljt-tI[(lnlt-t,j+f)sina(t,t,-,)ds,, L
b(t) =
ds
s{cos[a(t) - a(
(In Jt -
to\+ 1) +
+cos [a(to, t) + a(t,to)l} dso
L
Using the boundary
conditions
(3.1)
_!&(t)v(t)-b(t)o(t)]ds=G, L
and the notation
(3.12) here,
G=+dx41d~
we find (3.15)
L
We append the relationship (3.15) to the system (3.13) and a finite domain ((3.15) is not required in the case of an infinite Both equations (3.13) will be inhomogeneous in the case of where their right sides, f and h, respectively, without terms ak, flk willbe
(3.14) in the case of domain). an infinite domain s’, with the constants
918
A. L KalandU
f
(t) -= R (t) - d2uoi ds2, h (t) = Au,
( onL)
(3.16)
The function UO is defined by (2.12). Conditions (3.14) remain unchanged ex cept for one, for h- = 0, which evidently drops out here. Even in the case of a finite domain it is understood that a function analogous to UO can be introduced if only the plane problem for a (finite) simply-connected domain sot bounded by the contour Lo allows of a closed form solution. It is known that the plane problem of elasticity theory has been studied repeatedly by different authors by the method of integral equations [5 - 91. The advantage of the integral equations in v and o presented above is that their solution affords a possibility of determining the contour stresses of interest in problems for multiconnected domains, without any further calculations. A somewhat different system of equations for Y and o is constructed in [lo]. One of the equations of this system agrees exactly with the second equation in (3.13). As in [lo], the representation (3. 7) taking accamt of the boundary conditions of the problem in terms of the function w and resulting in the second equation in (3.13), is used above to construct the integral equations. Hence, it is sufficient to use just any one of the two possible conditions of the plane problem to obtain the complete system of integral equations. In this paper, the condition (I?,, ; as‘2 r i* on 1, is used
(f*, G
are functions
given on L ),while the condition CPU, a7’” = g* on L
(3.17)
is differentitation with respect to a certain fixed direcis given in [lo], where 8 / 3T tion coincident with the direction of the tangent at any point of the boundary L. The meaning of condition (3.17) is not completely clear to this writer: the initial conditions of the problem (3. 1) apparently do not permit the determination of the left side of (3.17) along the contour Lh. It is assumed R* z 0 in condition (3.17) on the outlines of holes free of external forces (only such holes were considered in [lo] ). Even if it is considered that such conditions are valid, it is not at all convenient to use them. They eliminate the without which the plane problem is generally incorrect; it is well constants ok, &, known that only a suitable selection of unknown constants a, /3 can assure the necessary single-valuedness of the elastic displacements. The lack of these constants in ~101 results in a certain number of linear algebraic equations for discrete values of the solution of the integral equations turning cut to be excess. However, it can happen in some particular cases that a part of the constants a, fi of the numerical results in the or even all the constants are zero. The acceptability examples of two circular holes considered in [lo] should apparently be explained by this (the constants a, fi are generally not needed in a problem with one hole). Furthermore, the first condition in (3.14) for k = 0 is absent in [lo]. and as will be seen later, this condition plays an essential part in the investigation of the integral their inequations (the authors of [lo] were not concerned with an investigation of tegral equations).
Plane problem of elsstIcity theory
919
4. Inve4tigetion of the integral crqurtfont, Letus start with the case of a finite domain and let us consider the homogeneous system of equations (4.1)
Let us introduce the function U biharmonic in ous functions V and CTon L by the integral
S+ , defined for arbitrary continu-
Let us call it ug for some solution vO? a, of the homogeneous system, and let us apply the Lapiace operator to it. We obtain
P The limit values of the right side of (4.3) on I. are zero when the point tends from s- to the point PO on L according to the first equation in (4.1). But the right side of (4.3) is harmonic in any of the domains S ,<- (k -_ 0, 1,. . , ! m) and is bounded at infinity because of the third equation in (4.1). Hence, on the basis of uniqueness of the harmonic function
After passing to the limit as P -+ PO from inside obtain, respectively, from (4.3) and (4.4)
and outside of
S’ I we
Let us fix the point PO on L , and let us differentiate (4.3) and (4.4) with respect to the direction of the external normal to L at the point PO. Passage to the limit in the equalities obtained in this manner as PO -+ P, we find
&
Avo(Po)= v. (PO)
and (4.2), whose left side equals
u.
(4.6)
for Y -+ Vo, o = o.
PO(P) =&~L"(vo;P*Q~d% L
on L
PES”
becomes
A. L Kalandiia
920
There hence follows on the basis of the Green’s formula (2.7)
-&
sl(c,,; P, Q)ds
= 0,
PES
(4.8)
L
~SIL”(l.O;P,Q)+Z(uu;P,Q)lds=
0,
PES-
L
Now, let us note that a biharmonic function in Sf , equal to the right side of (4.7), is continuous up to the boundary L . Its limit values have a second order derivative with respect to the arc of the contour, which equals zero identically on L because of the first equation in (4.1). This means that the function mentioned should take on constant values on L , and the second equality in (4.8) yields &@,,;P,Q)ds=
-&,
PELT,
k=O,i,...,
m
(4.9)
L
Limit values of the appropriate integral from S- on ,C are in the left side of (4.9), and hh are certain perfectly definite real constants. On the basis of the known properties of a double-layer potential, vO = hk on Lk (k = O,l,. . ., m) follows from the first equality in (4.8) and (4.9). The function expressed by the second integral in the left side of the second equality in (4.8) is harmonic in each of the domains S ,-comprising S-, and is bounded at infinity by virtue of the last equation in (4.1). Hence, (4.9) means
-i&(
00;
P, Q)ds = - At<,
P E Sk-,
k=O,l,...,m
(4.10)
L
Using the limit properties of a simple-layer potential exactly of (4.6), we find from the first equality in (4.8) and (4.9) au0 /an
=- 0
on
Now let us use the Green’s identity given in S+ by (4.7). We have
Lk
(k = 0, 1, . . ., m)
for the functions
1s (A,v,)*ds = 5 (A.v+-
s+
as in the derivation
- v,,$)
AuO, uo, where
ds
u.
is
(4.11)
L
Because of the equations established above for the limit values of the function uO and its normal derivative, as well as the relations written by the third equality in (4.1) for u = uo, the right side of the preceding formula equals zero. Therefore, A.v,=(I in S+,and vo(Q)=oo(Q)=O on Lonthebasisof(4.5)and (4.6). Uniqueness of the solution of the system (4.1) is proved. Turning to the case of an infinite domain, we first clarify the behavior of the we obtain the asympfunction (4.2) for large I 2 I. After elementary calculations, totic formula (A and B are real and complex constants) 8nv (P) = A ) z2 1In 12 ) + (BS f Bz) (In l 2 l + l/*) + (4.12)
O(ln
Iz I)
(Iz 1-m)
Flaw problem of elasticity lhcory
-
f v (t)ds,
A -
B =
i,
[
921
[tv (t) - it’a (t)] ds
i
In the case of the infinite domain, the homogeneous two equations in (4.1) obtained from the inhomogenecus = pk = 0 on Lk (k = 1, . ..,n), r = B' = additional equations to a total number of m -k 2: s v(t)&=0 %
(k=
(4.13)
system will consist of the first equations for fi = fa = ak C' = 0 and the following
(4.14)
1,...,7?1)
\ [xv (t) + y’o (t)] ds = [ [yv (t) i i
x’cr (t)] rls = 0
(L
is again the set of contours L,,L,,...,L,). Analogously to the preceding, we introduce the function ua defined in terms of the solution of the homogeneous system, by means of (4.2), and the limit uo, oo relationships (4.5) and (4.6) are established on the basis of the Green’s formula (2.21) for u. s 0 from the second equation in (4.1) and the first equations in (4.14). Using (2.20) for u. E (I and the first equation in (4.1) we find as in the case of a finite domain vo=-
h k,
dv,
I dn
=
0
where hk is some real constant Formula (4.11) remains for an infinite
on
Lk (k= 1,2,...,m)
domain.
In order to prove this, the integral (4.15)
Lk
is investigated for large 1z I. According to (4.14), the coefficients A and B from (4.. 13), which correspond to the function v0 , equal zero, and (4.12) yields the following estimate for u. v0 = 0 (in 17,I)
(4.16)
It shows that the function v. is representable outside La by (2. l), where ‘p aof_. On the basis of and x are the series (2.6) without the coefficients a0 and the above, the estimate (4.16) allows the following strengthening: ~Av, = 0 (12 I-“) A270= 0 (I z I-%), an = 0 (I z I-‘), vo = 0 (Q, + In the presence of the preceding formulas, the integral (4.15) tends to zero as I? -+OO and this indeed proves the validity of (4.11) in the case under consideration. As before, ~~~ = 0 in S+ follows, meaning y. (Q) = u. (0) = 0 on L. Let us turn to the inhomogeneous integral equations, and because of the complete Let us first analogy we limit ourselves to considering the case of a finite domain. note that the first equation in (3.14) can be written as
--$&PO) =0 0
(4.17)
A. I. Kalandiia
922
(k = 0, 1, . . . m) go (PO) - mz0 - BkY0 on L Limit values of the appropriate functions from the domain s+ are everywhere in the preceding equality. The function Q defined by (4.18), is single-valued and continuous on each of the contours Lh. for any v and G. Hence, the equation obtained from (4.17) by differentiation with respect to the arc so, namely, the equation )
(4.19)
da / ds,aQ (PO) := 0
is equivalent to the initial equation. (For the differentiation of (4.17) with respect to the arc so to be valid it is first necessary to raise the smoothness of the line L , to require, say, that the coordinates of its points 2, y have continuous derivatives with respect to S to the third order). Hence, if the first equation in (3.13) is replaced by (4.19) in the integral equations system under consideration, then we obtain a system completely equivalent to the initial system. On the other hand, the system (4.19) and the second equation in (3.13) will be a system of singular integral equations of normal type with zero index (see [4], Chapter VI), as can be seen on the basis of (3.13) for the kernel kij . As is known, the Fredholm theory which we shall use, is valid for such a system. The presence of the solution of the system (4.19) and the second equation in (3,13) is assured by the existence theorem for the solution of the fundamental biharmonic problem for multiconnected domains [5,9]. (We prefer not to resort to the appropriate Fredholm theorem here to avoid the investigation of the associated homogeneous system). The final number of arbitrary constants in the solution in terms of nontrivial solutions of the corresponding homogeneous system of integral equations is determined uniquely on the basis of the uniqueness, proved above, for the solution of the system (3.13), the first equation in (3.14) and (3.15). Hence, the solution U, o mentioned ak will depend exclusively on g, a k, I’,k and will contain the unknown constants b k linearly. If this solution is substituted into the last two conditions in (3.14), we then obtain which becomes a system of linear equations to determine sky Pk ( (4.20) in the notation “j
= dj,
pj 1 a,+j
(j z
1. 2. . . ., nt)
where Fi are unknown constants dependent on the functions which vanish for fi (t) = f2 (t) = 0 on L, and aik are dependent only on the geometry of the domain S+ (there is the integral equations in the system (4.20)). The system (4.20) defines the constants u ,+ (k = 1, 2, fact, let the system have the solution ako for Fk = 0 (k =
fi, f,
given on L and also definite constants no real need to solve . ,2m) uniquely. 1, 2,. . ., 2~).
In Then
Plane problem of elasticity theory
923
the system of integral equations (3.13), (3.14) will have a solution for I: (rj = o on L , and ak = aho, & = a;,,. Substituting this value of vO, csO in the right side of (4.2), we construct the functions u,, (P) which is biharmonic in S+ and satisfies the following boundary conditions on L
av, , -= av, Ti ay 8X
0
ah.’ + ia,,;,
on &.iL=O,
1, . . . . q:j)
and also the conditions of single-valuedness of the elastic displacements. Then u0 = ccnst in S- and all the zLkO= 0 on the basis of the uniqueness of the solution of the plane problem. Therefore, the system of integral equations (3.13) - (3.15) is solvable for any (sufficiently smooth) boundary values ii, fs subjected to the above-mentioned constraints, and determines the functions u, cr and the previously unknown constants akr pk uniquely. The integral equations(3.13)have kernels with logarithmic singularities and, as follows from the above, are equivalent in the sense of the existence of the solution of regular equations of the second kind. A numberical realization of the solutions can be accomplished by known, well-developed methods. REFERENCES 1.
Mus
khelishvili, N. I., Some Fundamental Problems of Mathematical Elasticity Theory, “Nauka”, Moscow, 1966. (See aIso,in mglish, Pergamon Press Vol. 2, No. 1, 1934.
2.
G r i o 1 i, G., Struttura della funzione di Airy nei sistemi essi. G. Matem. di Battaglini, Vol.77, 1947.
3.
of stress in an elastic solid, with Michell, J. H., On the direct determination applications to the theory of plates, Proc. London Math. Sot., Vol. 31, 1930.
4.
Muskhelishvili, 1962.
5.
M i c h 1 i n, S. B. a Le probl&me biharmonique C. R Acad. Sci. Vol. 197, No. 12, 1933.
6.
M u s k h e 1 i s h v i 1 i, N. I., New contour problems of plane elasticity No. 1, 1934.
7.
M u s k h e 1 i s h v i 1 i, N. I. Investigation of new integral equations elasticity theory, Dokl. Akad. Nauk SSSR, Vol.3, No. 2, 1934.
8.
S h e r m a n, D. I.,
9.
S h e r m a n, D. I.) Static plane problems Matem. Inst., Vol. 11, 1937.
N. I.,
Singular
Integral
Equations,
fondamental
molteplicement
Fizmatgiz,
conn-
Moscow,
2 deux dimensions,
general method of solving fundamental theory, Dokl. Akad. Nauk SSSR, Vol. 3,
of plane
On the solution of the plane static problem of elasticity theory for given external forces, DokI. Akad. Nauk SSSR, Vol. 28, No. 1, 1940. of elasticity
theory,
Trudy Tbilissk.
924
A. I. Kalandiia
10. C h r i s t i a n s e n, S. and H a n s e n, E.. A direct integral equation method for computing the hoop stress at holes in plane isotropic sheet, J. Qast., vol. 5, No. 1, 1975.
Translated by M. D. F.
PLANE PROBLEM OF PLASTICITY
THEORY XN A MULTICONNECTED
PMM Vol.43,
No.5, 1979, pp. 846-858 A. I. KALANDIIA (Tbilisi) (Received November 20, 1978)
Fredholm integral equations are constructed for unknown normal stresses on the domain boundary in the fundamental problem of plane elasticity theory, and their investigation is given. l. Formulation of the problem.Weshallsolvethefundamental plane problem of elasticity theory for external forces given on the boundary when the elastic medium occupies a finite or infinite domain of the plane of the complex variable z = 2 + iy, and the external forces on each of the bounding domains of the closed contours are statically equivalent to zero. We consider the (connected) domain S7 bounded by simple closed non-intersecting contours L,, L,,. . . , L,, of which the first encloses all the rest. The symbol L will denote the set of all contours. There may be no external contour L, , and then the domain s+ will be infinite (plane with holes). Let s- denote the domain complementing s+ in the whole plane, Sk- the domain enclosed within Lk (k = 1. 3.. . .’ in), and SD+, s,the domains inside and outside of L, , respectively. Furthermore, let n *be the normal to the line L drawn at any point and external relative to St, and T the direction of the positive tangent to - L at the same point. For definiteness, we consider that direction on I, positive which keeps the domain St on the right. We assume the line L to have continuous curvature everywhere. 2. On properties of stress functions in a multiconnected domain, Representation of biharmonic fun c t i o n s, To facilitate the presentation let us recall certain known formulas of plane elasticity theory and let us indicate the properties of the stress functions. We borrow all the necessary formulas from the book by N. I. Muskhelishvili [l]. The biharmonic stress function IL(7, g) (Airy function) and its first derivatives are representable by the formulas u = Re [Z(F (z) -I- y. (r.)], Z = .Z - iy (2.1) &L i)rr $ (2) -= j(’ (r.) ClX -+ i dy xzz‘p (3) + ZG) + 9(z), where cp, $, x are analytic functions of the variable 2 in a domain occupied by S+ they have the form (see [l], Sect. 35) an elastic medium. In the finite domain 9 (z) = z 5 k-1
A, In
(2.2) (
k=l
910
911
Plane problem of elastlclty theory
x (2) =
2
yA.’ In (2
i
k=l
‘II (‘1 = i
k=l
yk’ln
tz-
zk) + $,* tz)
where zk is a certain fixed point within Sk- (k = 1, 2,. . ., m), Ak is an arbitrary real number, ykr Yk’, YkH are arbitrary complex constants, and (Fe1 xer *,* are holomorphic (single- valued, analytic) functions in S +. It is seen that for the Airy function to be single-valued in S+ , it is necessary and sufficient to comply with the conditions Bk = yk’, Im yk” = 0 (k = 1, 2,. . .,
m)
(2.3)
and this means (see [l], Sect, 33) that the external forces on the contour of each of the holes are statically equivalent to zero. In the presence of (2.3), for the elastic displacements to be single-valued it is necessary (and sufficient) that Al, = 0, yk = 0 (k = 1, 2,.. ., m) and this is equivalent to the requirement of holomorphicity of ‘p (z) in latter conditions, expressed in terms of the function u (5, Y), are
s
f3AU ~cls=O,
’ St
6Au xan-Auan
cls=O
a2 > (k=l,2,
ds=O
S+. These
(2.4)
. . ..m)
S+ , Therefore, if the stress function u (z, Y) is single-valued in the domain then for the corresponding elastic displacements to be single-valued, it isnecessary and sufficient to comply with the conditions (2.4). ‘p (z), 9 (z) are holomorphic Upon compliance with these conditions, the functions x (z) will generally be multivalued (it has the form of the in SC, and the function right side in the second relationship of (2.2) for ok’ = 0, Im yh” = 0). The above-mentioned properties of the stress function were apparently first established by GrioIi [2] on the basis of the paper [S]. &, , Sf is an infinite domain located outIn the case when there is no contour In this case, under the same assumptions relative side the contours L,, L,,.. ., L,,. and the single-valuedness of the displacements, to the external forces applied to Lk , the nature of the stress functions as well as the potentials cp, x, ‘p remains as before in any finite subdomain of the domain S+. However, the functions cp and x will not generally be holomorphic in the whole domain S+. It is known that the Kolosov - Muskhelishvili potentials admit of the following representations in the neighborhood of the infinitely distant point (see [l], Sect. 36) (2.5) cp (z) = ‘Fo (z) + cp* (Z)? x (z) = X0 (z) + X* (2) (p. (z) = rz, x0 (z) = I”z2, I” = B' + iC'
A. I. Kalandiia
912
Here 1’, B’, C’ are real constants characterizing the given stress field at infinity, and 3. is a certain real constant. The formulas (2.6) mean that the functions ‘p* (z), are holomorphic outside a circle L, of sufficiently large radius R with 9* (z) center at the origin, including the infinitely distant point. Let us present certain fundamental formulas for representations of the biharmonic functions which are derivable by a standard method by using appropriate elementary solutions. A certain difficulity occurs just in the case of the infinite domains when the behavior of the function under consideration must be taken into account in infinitely remote parts of the plane. For the class of functions which we shall discuss later, this behavior is characterized completely by (2.5). The letters P, P, wiLl denote points of the domain S+ or S- , and Q, PO will denote points on the line I;, where Q will usually be the variable of integration. The affixes of these points on the complex plane shall be Z, zlr t, to, respectivz = .Z + iy, z1 = x1 + iy,. The arc abscissas of the points t, to are ely, where denoted by s and so and are measured in the direction of the positive tangent T to the line L on each contour L, (k = 0, 1, . . ., m) . We shall not introduce new symbols to denote the coordinates of the points Q and PO by assuming t = 5 -t iy, to = x0 -;- iy,, or more accurately, t = X(S) + my(s) and to = z (so) f iy (so). To avoid the introduction of new symbols for the functions f of points of the line L we shall also sometimes not distinguish the notation f (t) and f (Q) or f (to) and f (I’,). Writing the Green’s identity for the pair of functions u, Au, and then for Au, 0, adding the formulas obtained, and relying upon the elementary solution of the biharmonic equation, we obtain the known representations for a finite domain 1
5
P;
[LO (u;
u (a = -yjy
Q) -i1 (u; P; Q)] dsy
Z’ E ,j”
(2.7)
L
0 =
& s’ [Lo (u;
P;
Q) + 1 (n;
P;
PES-
Q)] ds,
L
Here the point
(a/an, 0): Pi
LO+; 1 (u; P;
Applying
is differentiation
Q) = 4 Q).+
1 [
r21n~
with respect to the normal direction
a
1 aAlL
--anq
QEL,
the Laplace operator to both sides of (2.7), P, Q) as,
L o=&
s
L
l(A.u; P, Q)ds,
PES-
at
I
A”anqr21nF
Q
& 11 (Au;
~0
1
+--l)~--ua$+,
A.u (P) =
to
r
= i5-tl
we obtain
P ES+
(2.8)
Plane problemof elasticity theory
The unction the condition
Au is evidently
harmonic
in
913
S+ , and should co~equently
satisfy
s
8Au YJ-+s=O
l
(2.9)
L
The known relationships
.(P)~ql(u:
s
o=$-
P. Q)ds,
PES-
P, Q)b,
l(u;
(2.10)
PEF
L
follow from (2.7) for a function u (x, Y) which is harmonic in S . Let us examine the case of an infinite domain in greater detail. Let us introduce a domain Sn of radius LR
L,, L,, . . ., ,!I+,, and within the circle is taken so large that the circle L, would enclose all the contours L,, k = 1, 2,. . ., m, whose set, i.e., total domain boundary, will again be denoted by L. Let us set located outside the contours R with center at the origin;
Uwhere
uo
R
(2.11)
ug = u+
is defined by the first formula in ( 2.1) and the last three formulas in (2.5) uO = Re [Z cpo(4 +
x0
(41
(2.12)
=
ps [r --f-B’ cos 26 - C’ sin 2 61, z = Wie Gxfm) = 2 (f - B’), q$(=) = 2 (r + B’), TXJrn) = 2 C’ (a,, uV, zxV are the stress components). components, i. e., the quantities (2.13),
(2.13)
Let us recall that the values of the stress
are given at infinity when considering the plane problem for domains containing the infinitely distant point of the plane. The right side of (2,12) is an Airy function corresponding to the homogeneous stress field caused by the forces (2.13). It is biharmonic in any finite part of the plane and for large 1z 1 ug = a (p*) (2.14) The relationships (2.7) are evidently valid in S,
for the function
1~-uo . There-
fore 2 u* (P) = 2n
s
[r,” (u,;
P,
Q, + 2 (u*:
P,
Q)] ds,
P ES,
L+Lf$ 0
1 =-zi-
s L+LR
[L” (u*;
P,
Q) + 1 (IA,; P,
Q)]ds,
P E
Sk-,
k = 1,x,
... , m
Moreover, analogously to (2.9)
s
L-tLR
Let us consider the function
aAu, an
(2.15) ds=O
A. L Kalandiia
914
(2.16) 1 I(Au,;P, Q)ds
Ax0 (P) = -&
“k
where P is an arbitrary point in any finite part of the plane first formula in (2.6), the estimates
i-‘Au,
Au = 0 (I z I-“), are valid for large I z ‘I s consequently It hence follows that
in the limit
as
that upon com$ance
cannot grow more rapidly than plnp hence conclude that for large 1z )
on the plane
z
30.
(2.17)
(2.15) becomes
’ r3Air a,2 as = 0
(2.18)
with the condition
=
tends to zero at H -
\ i,
at infinity.
Lao
Now on the basis of (2.17)
in (2.16)
XI , the equality
ds =
on the basis of the
0 (I I” I-3)
0 everywhere
R -
- rlh, 7 s O/I I, It is almost evident
=
the integral
A(/%,On the same basis,
an
z.
!I
(2.18)
the function
Taking (2.14) into account,
(I”“)
we
(2.19)
and (2.19) 00 (I. Y) = 31 + BY + Y
follows, where a, B, Y are arbitrary real constants. this trinomial since it does not yield nonzero stresses. In the long run, the representations 1 ’ 7~(P) = u,, (PI -kg-
Q) T I (I!.; P,
Q)] ds,
[L” (u; I’, Q) + 1 (!I; f’, Q)] ds,
PfSh_-,
s
[L” (1~; P,
We shall not turn attention
to
(2.20)
P E S+
L ~0
(p) =
-&,\
h-=1,2,
,..
, m
L
hold for the stress function Correspondingly
u (x, y) in the infinite 1
domain
S+
(2.21)
Au0 (Pj = -&
1 1 (Au; P, L
Q) ds,
PESk-,
k=l,2,
.... m
Planeproblem of elasllcity theory
915
Now (2.18) should still be added to the preceding formulas, It is understood the conditions (2.4) that the displacements be single- valued remain unchanged. 3. Integral equations of the conditions introduced, the plane problem generalized plane state of stress of an elastic stress function u (x, y) in the domain S+ g+
ig
that
the plane problem. Under (the problem of plane strain or the body) is to determine the biharmonic by means of the boundary conditions
=fl(t)+if2(t)fc(t)
on
L
(3.1)
where fl, f2 are functions given on L which are single-valued and continuous each of the contours Lk comprising L , and subject to the conditions:
s
flds+f:!dy=O
on
(k=1,2,...,7?2)
(3.2)
and c (1) - ck = ak + iflk on Lk, where ck are complex constants to be determined together with the function u . Only one of the constants ck can be fixed arbitrarily; we shall consider c (t) ‘= 0 on Lo . Moreover,. it is required from the function u that it satisfy the following additional conditions
s.
J$is=O
(k=
0,1,2,...,m)
(3.3)
9;
SC
a$-Au$)ds=O,
~(y~-Au~)&~O
LI;
(3.4)
Lk
(k = 1,2, . . . ) m)
In the case of an infinite domain, the boundary conditions (3.1) are given in the set L,, L?,. . ., L,, and in conformity with this, one of the additional conditions, namely (3.3) ) drops out for k = 0 . The condition at infinity u(z,
Y) = UO(T,
Y) iO(lzl)
(3.5)
should still be appended to these conditions, where ~0 is given by (2.12). and the second term is the right side of the first formula in (2.1) if the corresponding right all the second sides of (2.6) are substituted in place of cp and x . (More accurately, order partial derivatives of u should be given at infinity, and this wilI be equivalent to giving the function ?.Aitself in the form (3.5) ). The COIBtant dk can here alS0 be given arbitrarily on any one of the contours Lk (k = 1, 2,. . . , m), be considered zero, say, as we shall do. Therefore, the number of unknown real constants to be determined during the solution of the problem equals 2m and 2m - 2 in the case of the finite and infinite domains, respectively. proceeding to the construction of the integral equations for the finite domain S’J we rewrite condition (3.1) in the form ~=go(t)+%z-!-BkY+~k
=s
(3.6)
s
l
go(t)
j&x t- fz4/,
0
b(t)
=
f,$ -+j,:
on Lk(k = 0, I,..., m)
916
A. L Kalandiia
According to the above, it can be considered neglect the constants aR since they play no part Let us insert (3.6) in place of 11 and &di3n ity in (2.7), and let us take into account that the everywhere harmonic. Then by using (2.10) we entation for the solution of the problem formulated ted in the right side): (A(P),
-&\
?U(P)-=
&S
’ L (u; P; Q)ds
that o,,, =_ f10 mmOY and we shall in solving the problem. in the right side of the first equalfunction ahx -t- Fky + hk is obtain the following integral repres(the additive constant X0 is omit-
‘; w(P),
PES’
(3.7)
[(lllf-l)h”~V--p,(Q)~~,lf]ds
(3.8)
The function zu (P) (the sum of simple and double layer potentials with known densities) is given on the whole plane. It undergoes a discontinuity when the point P passes through the line L t which is not essential to the subsequent exposition. Let us pass to the limit P -+ P o (PO E L) in (3.7), and let us differentiate the limit equality twice with respect to the contour arc. (For the differentiation with respect to the arc s to be legitimate, it is necessary that the given functions go and h, be subject to definite smootheness conditions which are easily refined). Then using the boundary condition (3.6) for the function u (P) desired, we find (the prime denotes differentiation with respect to s)
-s’ -f$ 1
2x
L” (u; PO, Q) ds = g (to) + QX~ + pkq/
on
L,,
(3.9)
L
(k = 0, 1,2:.
. . , m;
g (to) = g,” (to) -
t” = .ql -.I- iy,) a2w (PO) 1 &2
(3.10)
We also pass to the limit as P 3 PO in the first equality in (2.8) and we use Then the known formula for the limit values of a double-layer potential. u; Pm Q)ds,
Au(P,,)=$~E(A
P,, E
L
(3.11)
L
where the limit values on the line L for the function Au are in the left side and values of the integral in the first equality in (2.8) on the same line are in the right side, The set of equations (3.9X (3.11) yields a system of integral equations in the unknown boundary values of the functions AU and dAu / dn. After the calculaJ_,” under the integral sign in (3.9), tions related to differentiation of the operator by using the notation Au(Q) we write the system of integral &
=; o CC% equations
+Au
= v(Q)
in the form
1 [kir (to, t) v (r) + k,, (to, r) o (t)] ds = g (to) + or&” + l%vo” L th(to, s
qv
(t) -t
(3.12)
h27,(to, 0 CJ@>I d.9 = 0
(3.13)
Plane problem of elrutldty
-
4k,, (to, t) = 2 (hl r + 1) + k (to)
-
COS
917
theory
2a (t, t,,) -
r (21n F
+
i)X
sin a (t, tJ
2k,, (to, t) = r-l sin a (t,, t) - r-l sin 2a (t, t,) cos a (to, t) k (to) sin a (t, to) sin a (to, t) + k (to) cos ]a (t) - a (t,)l x
(In r + l/J kzl (to, t) = 2 (In F + I),
kaz(to, t) = 2 $--In
+
=
Q 2r-‘sin
a (to, t)
(F =
1t -
to ( )
where k (t) is the curvature of the line L at the point t, a (t) is the angle formOT, ed by the (positive) tangent to the line L at the point t and the axis and a (to, t) is also an angle formed by the same tangent and the vector t,t , and a (t,) and a (t, t,) measured in the positive direction from this latter. The notation has an analogous meaning (the angles a (t,, t), and a (t, to) are shown graphically with their measurement directions in [4] ). The additional conditions (3.3) and (3.4), which become
[v(t)ds=O
(3.14)
(k=O,l,...,m)
L;, ~[XV(t)+y’~(t)]ds=o, ;;=
S[yv(t)-.&o(t)]&-0 5
l,Z,....m)
in the new notation, are appended to (3.13). Let us mention still another relationship which the functions Y and o should satisty. To this end, we calculate the normal derivative of the desired function u (z, y) L . Taking into account that by means of (3.7) and we integrate it along the line the function 11)is harmonic in Sf , we find
s
&s
;ds=
I(t)=
b(t) Au
[a(t)%-
I
ljt-tI[(lnlt-t,j+f)sina(t,t,-,)ds,, L
b(t) =
ds
s{cos[a(t) - a(
(In Jt -
to\+ 1) +
+cos [a(to, t) + a(t,to)l} dso
L
Using the boundary
conditions
(3.1)
_!&(t)v(t)-b(t)o(t)]ds=G, L
and the notation
(3.12) here,
G=+dx41d~
we find (3.15)
L
We append the relationship (3.15) to the system (3.13) and a finite domain ((3.15) is not required in the case of an infinite Both equations (3.13) will be inhomogeneous in the case of where their right sides, f and h, respectively, without terms ak, flk willbe
(3.14) in the case of domain). an infinite domain s’, with the constants
918
A. L KalandU
f
(t) -= R (t) - d2uoi ds2, h (t) = Au,
( onL)
(3.16)
The function UO is defined by (2.12). Conditions (3.14) remain unchanged ex cept for one, for h- = 0, which evidently drops out here. Even in the case of a finite domain it is understood that a function analogous to UO can be introduced if only the plane problem for a (finite) simply-connected domain sot bounded by the contour Lo allows of a closed form solution. It is known that the plane problem of elasticity theory has been studied repeatedly by different authors by the method of integral equations [5 - 91. The advantage of the integral equations in v and o presented above is that their solution affords a possibility of determining the contour stresses of interest in problems for multiconnected domains, without any further calculations. A somewhat different system of equations for Y and o is constructed in [lo]. One of the equations of this system agrees exactly with the second equation in (3.13). As in [lo], the representation (3. 7) taking accamt of the boundary conditions of the problem in terms of the function w and resulting in the second equation in (3.13), is used above to construct the integral equations. Hence, it is sufficient to use just any one of the two possible conditions of the plane problem to obtain the complete system of integral equations. In this paper, the condition (I?,, ; as‘2 r i* on 1, is used
(f*, G
are functions
given on L ),while the condition CPU, a7’” = g* on L
(3.17)
is differentitation with respect to a certain fixed direcis given in [lo], where 8 / 3T tion coincident with the direction of the tangent at any point of the boundary L. The meaning of condition (3.17) is not completely clear to this writer: the initial conditions of the problem (3. 1) apparently do not permit the determination of the left side of (3.17) along the contour Lh. It is assumed R* z 0 in condition (3.17) on the outlines of holes free of external forces (only such holes were considered in [lo] ). Even if it is considered that such conditions are valid, it is not at all convenient to use them. They eliminate the without which the plane problem is generally incorrect; it is well constants ok, &, known that only a suitable selection of unknown constants a, /3 can assure the necessary single-valuedness of the elastic displacements. The lack of these constants in ~101 results in a certain number of linear algebraic equations for discrete values of the solution of the integral equations turning cut to be excess. However, it can happen in some particular cases that a part of the constants a, fi of the numerical results in the or even all the constants are zero. The acceptability examples of two circular holes considered in [lo] should apparently be explained by this (the constants a, fi are generally not needed in a problem with one hole). Furthermore, the first condition in (3.14) for k = 0 is absent in [lo]. and as will be seen later, this condition plays an essential part in the investigation of the integral their inequations (the authors of [lo] were not concerned with an investigation of tegral equations).
Plane problem of elsstIcity theory
919
4. Inve4tigetion of the integral crqurtfont, Letus start with the case of a finite domain and let us consider the homogeneous system of equations (4.1)
Let us introduce the function U biharmonic in ous functions V and CTon L by the integral
S+ , defined for arbitrary continu-
Let us call it ug for some solution vO? a, of the homogeneous system, and let us apply the Lapiace operator to it. We obtain
P The limit values of the right side of (4.3) on I. are zero when the point tends from s- to the point PO on L according to the first equation in (4.1). But the right side of (4.3) is harmonic in any of the domains S ,<- (k -_ 0, 1,. . , ! m) and is bounded at infinity because of the third equation in (4.1). Hence, on the basis of uniqueness of the harmonic function
After passing to the limit as P -+ PO from inside obtain, respectively, from (4.3) and (4.4)
and outside of
S’ I we
Let us fix the point PO on L , and let us differentiate (4.3) and (4.4) with respect to the direction of the external normal to L at the point PO. Passage to the limit in the equalities obtained in this manner as PO -+ P, we find
&
Avo(Po)= v. (PO)
and (4.2), whose left side equals
u.
(4.6)
for Y -+ Vo, o = o.
PO(P) =&~L"(vo;P*Q~d% L
on L
PES”
becomes
A. L Kalandiia
920
There hence follows on the basis of the Green’s formula (2.7)
-&
sl(c,,; P, Q)ds
= 0,
PES
(4.8)
L
~SIL”(l.O;P,Q)+Z(uu;P,Q)lds=
0,
PES-
L
Now, let us note that a biharmonic function in Sf , equal to the right side of (4.7), is continuous up to the boundary L . Its limit values have a second order derivative with respect to the arc of the contour, which equals zero identically on L because of the first equation in (4.1). This means that the function mentioned should take on constant values on L , and the second equality in (4.8) yields &@,,;P,Q)ds=
-&,
PELT,
k=O,i,...,
m
(4.9)
L
Limit values of the appropriate integral from S- on ,C are in the left side of (4.9), and hh are certain perfectly definite real constants. On the basis of the known properties of a double-layer potential, vO = hk on Lk (k = O,l,. . ., m) follows from the first equality in (4.8) and (4.9). The function expressed by the second integral in the left side of the second equality in (4.8) is harmonic in each of the domains S ,-comprising S-, and is bounded at infinity by virtue of the last equation in (4.1). Hence, (4.9) means
-i&(
00;
P, Q)ds = - At<,
P E Sk-,
k=O,l,...,m
(4.10)
L
Using the limit properties of a simple-layer potential exactly of (4.6), we find from the first equality in (4.8) and (4.9) au0 /an
=- 0
on
Now let us use the Green’s identity given in S+ by (4.7). We have
Lk
(k = 0, 1, . . ., m)
for the functions
1s (A,v,)*ds = 5 (A.v+-
s+
as in the derivation
- v,,$)
AuO, uo, where
ds
u.
is
(4.11)
L
Because of the equations established above for the limit values of the function uO and its normal derivative, as well as the relations written by the third equality in (4.1) for u = uo, the right side of the preceding formula equals zero. Therefore, A.v,=(I in S+,and vo(Q)=oo(Q)=O on Lonthebasisof(4.5)and (4.6). Uniqueness of the solution of the system (4.1) is proved. Turning to the case of an infinite domain, we first clarify the behavior of the we obtain the asympfunction (4.2) for large I 2 I. After elementary calculations, totic formula (A and B are real and complex constants) 8nv (P) = A ) z2 1In 12 ) + (BS f Bz) (In l 2 l + l/*) + (4.12)
O(ln
Iz I)
(Iz 1-m)
Flaw problem of elasticity lhcory
-
f v (t)ds,
A -
B =
i,
[
921
[tv (t) - it’a (t)] ds
i
In the case of the infinite domain, the homogeneous two equations in (4.1) obtained from the inhomogenecus = pk = 0 on Lk (k = 1, . ..,n), r = B' = additional equations to a total number of m -k 2: s v(t)&=0 %
(k=
(4.13)
system will consist of the first equations for fi = fa = ak C' = 0 and the following
(4.14)
1,...,7?1)
\ [xv (t) + y’o (t)] ds = [ [yv (t) i i
x’cr (t)] rls = 0
(L
is again the set of contours L,,L,,...,L,). Analogously to the preceding, we introduce the function ua defined in terms of the solution of the homogeneous system, by means of (4.2), and the limit uo, oo relationships (4.5) and (4.6) are established on the basis of the Green’s formula (2.21) for u. s 0 from the second equation in (4.1) and the first equations in (4.14). Using (2.20) for u. E (I and the first equation in (4.1) we find as in the case of a finite domain vo=-
h k,
dv,
I dn
=
0
where hk is some real constant Formula (4.11) remains for an infinite
on
Lk (k= 1,2,...,m)
domain.
In order to prove this, the integral (4.15)
Lk
is investigated for large 1z I. According to (4.14), the coefficients A and B from (4.. 13), which correspond to the function v0 , equal zero, and (4.12) yields the following estimate for u. v0 = 0 (in 17,I)
(4.16)
It shows that the function v. is representable outside La by (2. l), where ‘p aof_. On the basis of and x are the series (2.6) without the coefficients a0 and the above, the estimate (4.16) allows the following strengthening: ~Av, = 0 (12 I-“) A270= 0 (I z I-%), an = 0 (I z I-‘), vo = 0 (Q, + In the presence of the preceding formulas, the integral (4.15) tends to zero as I? -+OO and this indeed proves the validity of (4.11) in the case under consideration. As before, ~~~ = 0 in S+ follows, meaning y. (Q) = u. (0) = 0 on L. Let us turn to the inhomogeneous integral equations, and because of the complete Let us first analogy we limit ourselves to considering the case of a finite domain. note that the first equation in (3.14) can be written as
--$&PO) =0 0
(4.17)
A. I. Kalandiia
922
(k = 0, 1, . . . m) go (PO) - mz0 - BkY0 on L Limit values of the appropriate functions from the domain s+ are everywhere in the preceding equality. The function Q defined by (4.18), is single-valued and continuous on each of the contours Lh. for any v and G. Hence, the equation obtained from (4.17) by differentiation with respect to the arc so, namely, the equation )
(4.19)
da / ds,aQ (PO) := 0
is equivalent to the initial equation. (For the differentiation of (4.17) with respect to the arc so to be valid it is first necessary to raise the smoothness of the line L , to require, say, that the coordinates of its points 2, y have continuous derivatives with respect to S to the third order). Hence, if the first equation in (3.13) is replaced by (4.19) in the integral equations system under consideration, then we obtain a system completely equivalent to the initial system. On the other hand, the system (4.19) and the second equation in (3.13) will be a system of singular integral equations of normal type with zero index (see [4], Chapter VI), as can be seen on the basis of (3.13) for the kernel kij . As is known, the Fredholm theory which we shall use, is valid for such a system. The presence of the solution of the system (4.19) and the second equation in (3,13) is assured by the existence theorem for the solution of the fundamental biharmonic problem for multiconnected domains [5,9]. (We prefer not to resort to the appropriate Fredholm theorem here to avoid the investigation of the associated homogeneous system). The final number of arbitrary constants in the solution in terms of nontrivial solutions of the corresponding homogeneous system of integral equations is determined uniquely on the basis of the uniqueness, proved above, for the solution of the system (3.13), the first equation in (3.14) and (3.15). Hence, the solution U, o mentioned ak will depend exclusively on g, a k, I’,k and will contain the unknown constants b k linearly. If this solution is substituted into the last two conditions in (3.14), we then obtain which becomes a system of linear equations to determine sky Pk ( (4.20) in the notation “j
= dj,
pj 1 a,+j
(j z
1. 2. . . ., nt)
where Fi are unknown constants dependent on the functions which vanish for fi (t) = f2 (t) = 0 on L, and aik are dependent only on the geometry of the domain S+ (there is the integral equations in the system (4.20)). The system (4.20) defines the constants u ,+ (k = 1, 2, fact, let the system have the solution ako for Fk = 0 (k =
fi, f,
given on L and also definite constants no real need to solve . ,2m) uniquely. 1, 2,. . ., 2~).
In Then
Plane problem of elasticity theory
923
the system of integral equations (3.13), (3.14) will have a solution for I: (rj = o on L , and ak = aho, & = a;,,. Substituting this value of vO, csO in the right side of (4.2), we construct the functions u,, (P) which is biharmonic in S+ and satisfies the following boundary conditions on L
av, , -= av, Ti ay 8X
0
ah.’ + ia,,;,
on &.iL=O,
1, . . . . q:j)
and also the conditions of single-valuedness of the elastic displacements. Then u0 = ccnst in S- and all the zLkO= 0 on the basis of the uniqueness of the solution of the plane problem. Therefore, the system of integral equations (3.13) - (3.15) is solvable for any (sufficiently smooth) boundary values ii, fs subjected to the above-mentioned constraints, and determines the functions u, cr and the previously unknown constants akr pk uniquely. The integral equations(3.13)have kernels with logarithmic singularities and, as follows from the above, are equivalent in the sense of the existence of the solution of regular equations of the second kind. A numberical realization of the solutions can be accomplished by known, well-developed methods. REFERENCES 1.
Mus
khelishvili, N. I., Some Fundamental Problems of Mathematical Elasticity Theory, “Nauka”, Moscow, 1966. (See aIso,in mglish, Pergamon Press Vol. 2, No. 1, 1934.
2.
G r i o 1 i, G., Struttura della funzione di Airy nei sistemi essi. G. Matem. di Battaglini, Vol.77, 1947.
3.
of stress in an elastic solid, with Michell, J. H., On the direct determination applications to the theory of plates, Proc. London Math. Sot., Vol. 31, 1930.
4.
Muskhelishvili, 1962.
5.
M i c h 1 i n, S. B. a Le probl&me biharmonique C. R Acad. Sci. Vol. 197, No. 12, 1933.
6.
M u s k h e 1 i s h v i 1 i, N. I., New contour problems of plane elasticity No. 1, 1934.
7.
M u s k h e 1 i s h v i 1 i, N. I. Investigation of new integral equations elasticity theory, Dokl. Akad. Nauk SSSR, Vol.3, No. 2, 1934.
8.
S h e r m a n, D. I.,
9.
S h e r m a n, D. I.) Static plane problems Matem. Inst., Vol. 11, 1937.
N. I.,
Singular
Integral
Equations,
fondamental
molteplicement
Fizmatgiz,
conn-
Moscow,
2 deux dimensions,
general method of solving fundamental theory, Dokl. Akad. Nauk SSSR, Vol. 3,
of plane
On the solution of the plane static problem of elasticity theory for given external forces, DokI. Akad. Nauk SSSR, Vol. 28, No. 1, 1940. of elasticity
theory,
Trudy Tbilissk.
924
A. I. Kalandiia
10. C h r i s t i a n s e n, S. and H a n s e n, E.. A direct integral equation method for computing the hoop stress at holes in plane isotropic sheet, J. Qast., vol. 5, No. 1, 1975.
Translated by M. D. F.